# Properties

 Label 4025.2.a.r Level 4025 Weight 2 Character orbit 4025.a Self dual yes Analytic conductor 32.140 Analytic rank 0 Dimension 5 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4025 = 5^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1397868136$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.122821.1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 805) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( 1 + \beta_{3} ) q^{3} + ( \beta_{3} + \beta_{4} ) q^{4} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{6} + q^{7} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{8} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( 1 + \beta_{3} ) q^{3} + ( \beta_{3} + \beta_{4} ) q^{4} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{6} + q^{7} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{8} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{9} + ( -2 + 2 \beta_{2} - \beta_{3} ) q^{11} + ( 2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{12} + ( \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{13} + ( 1 - \beta_{1} ) q^{14} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{16} + ( 2 - 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} ) q^{17} + ( 3 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{18} + ( 2 - 3 \beta_{1} - \beta_{4} ) q^{19} + ( 1 + \beta_{3} ) q^{21} + ( -1 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{22} - q^{23} + ( 1 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{24} + ( 3 - 4 \beta_{1} - \beta_{3} + \beta_{4} ) q^{26} + ( 4 - 4 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{27} + ( \beta_{3} + \beta_{4} ) q^{28} + ( -2 + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{29} + ( -2 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{31} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{32} + ( -3 + \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{4} ) q^{33} + ( -1 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{34} + ( 5 - 4 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} + 3 \beta_{4} ) q^{36} + ( 2 \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{37} + ( 4 + 2 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} ) q^{38} + ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{39} + ( -1 + 4 \beta_{1} + 2 \beta_{4} ) q^{41} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{42} + ( 1 + 4 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} ) q^{43} + ( \beta_{1} + 2 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} ) q^{44} + ( -1 + \beta_{1} ) q^{46} + ( 6 - \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{47} + ( 2 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{48} + q^{49} + ( -3 - \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{51} + ( 7 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{52} + ( 3 + \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{53} + ( 7 - 2 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} + 5 \beta_{4} ) q^{54} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{56} + ( 6 - 6 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{57} + ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{58} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{59} + ( 1 + \beta_{1} + 3 \beta_{2} ) q^{61} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{62} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{63} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{64} + ( -10 + 9 \beta_{1} + 6 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} ) q^{66} + ( 6 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{67} + ( -8 + \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{68} + ( -1 - \beta_{3} ) q^{69} + ( -10 + 5 \beta_{1} + \beta_{2} + 4 \beta_{3} + 2 \beta_{4} ) q^{71} + ( 6 - 5 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 5 \beta_{4} ) q^{72} + ( 1 + 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{73} + ( 1 - \beta_{1} + 3 \beta_{2} - 5 \beta_{3} - 4 \beta_{4} ) q^{74} + ( 3 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{76} + ( -2 + 2 \beta_{2} - \beta_{3} ) q^{77} + ( 3 - 7 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{78} + ( 1 + 5 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{79} + ( 4 - 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} ) q^{81} + ( -3 - 5 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} ) q^{82} + ( 2 - 3 \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} ) q^{83} + ( 2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{84} + ( -5 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - 5 \beta_{4} ) q^{86} + ( 8 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} - 7 \beta_{4} ) q^{87} + ( -7 + 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{88} + ( -2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} ) q^{89} + ( \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{91} + ( -\beta_{3} - \beta_{4} ) q^{92} + ( 4 - 4 \beta_{1} - 5 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{93} + ( 4 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{94} + ( 5 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{96} + ( 1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 5 \beta_{4} ) q^{97} + ( 1 - \beta_{1} ) q^{98} + ( -10 + 7 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} - 5 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 3q^{2} + 6q^{3} + 3q^{4} + 7q^{6} + 5q^{7} + 3q^{8} + 5q^{9} + O(q^{10})$$ $$5q + 3q^{2} + 6q^{3} + 3q^{4} + 7q^{6} + 5q^{7} + 3q^{8} + 5q^{9} - 11q^{11} + 14q^{12} + 7q^{13} + 3q^{14} - q^{16} - q^{17} + 17q^{18} + 2q^{19} + 6q^{21} - 2q^{22} - 5q^{23} + 12q^{24} + 8q^{26} + 15q^{27} + 3q^{28} - 14q^{29} - 6q^{31} + 4q^{32} - 22q^{33} + 4q^{34} + 28q^{36} + q^{37} + 31q^{38} + 15q^{39} + 7q^{41} + 7q^{42} + 12q^{43} - 11q^{44} - 3q^{46} + 24q^{47} + 4q^{48} + 5q^{49} - 28q^{51} + 36q^{52} + 21q^{53} + 49q^{54} + 3q^{56} + 15q^{57} - 9q^{58} - q^{59} + 7q^{61} + 26q^{62} + 5q^{63} - 15q^{64} - 45q^{66} + 35q^{67} - 43q^{68} - 6q^{69} - 32q^{71} + 36q^{72} + 7q^{73} - 10q^{74} + 10q^{76} - 11q^{77} + 8q^{78} + 18q^{79} + 21q^{81} - 33q^{82} + q^{83} + 14q^{84} - 26q^{86} + 18q^{87} - 41q^{88} + q^{89} + 7q^{91} - 3q^{92} + 19q^{93} + 14q^{94} + 26q^{96} + 9q^{97} + 3q^{98} - 54q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 4 x^{3} + 4 x^{2} + 3 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 3 \nu + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 3 \nu^{2} + 2 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + 2 \nu^{3} + 4 \nu^{2} - 4 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + 2 \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 7 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 18 \beta_{1} + 2$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.79802 1.17316 0.266708 −0.787589 −1.45030
−1.79802 1.59027 1.23287 0 −2.85933 1.00000 1.37931 −0.471046 0
1.2 −0.173158 −1.11762 −1.97002 0 0.193524 1.00000 0.687440 −1.75093 0
1.3 0.733292 2.28713 −1.46228 0 1.67714 1.00000 −2.53886 2.23098 0
1.4 1.78759 −0.0742225 1.19548 0 −0.132679 1.00000 −1.43816 −2.99449 0
1.5 2.45030 3.31444 4.00395 0 8.12135 1.00000 4.91027 7.98549 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4025.2.a.r 5
5.b even 2 1 805.2.a.k 5
15.d odd 2 1 7245.2.a.bi 5
35.c odd 2 1 5635.2.a.x 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.k 5 5.b even 2 1
4025.2.a.r 5 1.a even 1 1 trivial
5635.2.a.x 5 35.c odd 2 1
7245.2.a.bi 5 15.d odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$-1$$
$$23$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{5} - 3 T_{2}^{4} - 2 T_{2}^{3} + 10 T_{2}^{2} - 4 T_{2} - 1$$ $$T_{3}^{5} - 6 T_{3}^{4} + 8 T_{3}^{3} + 7 T_{3}^{2} - 13 T_{3} - 1$$ $$T_{11}^{5} + 11 T_{11}^{4} + 14 T_{11}^{3} - 185 T_{11}^{2} - 612 T_{11} - 452$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 8 T^{2} - 14 T^{3} + 24 T^{4} - 33 T^{5} + 48 T^{6} - 56 T^{7} + 64 T^{8} - 48 T^{9} + 32 T^{10}$$
$3$ $$1 - 6 T + 23 T^{2} - 65 T^{3} + 149 T^{4} - 283 T^{5} + 447 T^{6} - 585 T^{7} + 621 T^{8} - 486 T^{9} + 243 T^{10}$$
$5$ 1
$7$ $$( 1 - T )^{5}$$
$11$ $$1 + 11 T + 69 T^{2} + 299 T^{3} + 1060 T^{4} + 3464 T^{5} + 11660 T^{6} + 36179 T^{7} + 91839 T^{8} + 161051 T^{9} + 161051 T^{10}$$
$13$ $$1 - 7 T + 50 T^{2} - 218 T^{3} + 1149 T^{4} - 4063 T^{5} + 14937 T^{6} - 36842 T^{7} + 109850 T^{8} - 199927 T^{9} + 371293 T^{10}$$
$17$ $$1 + T + 25 T^{2} + 83 T^{3} + 704 T^{4} + 800 T^{5} + 11968 T^{6} + 23987 T^{7} + 122825 T^{8} + 83521 T^{9} + 1419857 T^{10}$$
$19$ $$1 - 2 T + 50 T^{2} + T^{3} + 1137 T^{4} + 1030 T^{5} + 21603 T^{6} + 361 T^{7} + 342950 T^{8} - 260642 T^{9} + 2476099 T^{10}$$
$23$ $$( 1 + T )^{5}$$
$29$ $$1 + 14 T + 106 T^{2} + 324 T^{3} - 723 T^{4} - 12204 T^{5} - 20967 T^{6} + 272484 T^{7} + 2585234 T^{8} + 9901934 T^{9} + 20511149 T^{10}$$
$31$ $$1 + 6 T + 85 T^{2} + 479 T^{3} + 3931 T^{4} + 20655 T^{5} + 121861 T^{6} + 460319 T^{7} + 2532235 T^{8} + 5541126 T^{9} + 28629151 T^{10}$$
$37$ $$1 - T + 17 T^{2} + 11 T^{3} + 376 T^{4} - 10076 T^{5} + 13912 T^{6} + 15059 T^{7} + 861101 T^{8} - 1874161 T^{9} + 69343957 T^{10}$$
$41$ $$1 - 7 T + 135 T^{2} - 1002 T^{3} + 9005 T^{4} - 58961 T^{5} + 369205 T^{6} - 1684362 T^{7} + 9304335 T^{8} - 19780327 T^{9} + 115856201 T^{10}$$
$43$ $$1 - 12 T + 138 T^{2} - 1233 T^{3} + 10559 T^{4} - 69898 T^{5} + 454037 T^{6} - 2279817 T^{7} + 10971966 T^{8} - 41025612 T^{9} + 147008443 T^{10}$$
$47$ $$1 - 24 T + 431 T^{2} - 5125 T^{3} + 50209 T^{4} - 375429 T^{5} + 2359823 T^{6} - 11321125 T^{7} + 44747713 T^{8} - 117112344 T^{9} + 229345007 T^{10}$$
$53$ $$1 - 21 T + 407 T^{2} - 4757 T^{3} + 50566 T^{4} - 385716 T^{5} + 2679998 T^{6} - 13362413 T^{7} + 60592939 T^{8} - 165700101 T^{9} + 418195493 T^{10}$$
$59$ $$1 + T + 225 T^{2} + 105 T^{3} + 22792 T^{4} + 5604 T^{5} + 1344728 T^{6} + 365505 T^{7} + 46210275 T^{8} + 12117361 T^{9} + 714924299 T^{10}$$
$61$ $$1 - 7 T + 259 T^{2} - 1417 T^{3} + 29142 T^{4} - 123032 T^{5} + 1777662 T^{6} - 5272657 T^{7} + 58788079 T^{8} - 96920887 T^{9} + 844596301 T^{10}$$
$67$ $$1 - 35 T + 785 T^{2} - 11891 T^{3} + 140386 T^{4} - 1278568 T^{5} + 9405862 T^{6} - 53378699 T^{7} + 236098955 T^{8} - 705289235 T^{9} + 1350125107 T^{10}$$
$71$ $$1 + 32 T + 619 T^{2} + 8265 T^{3} + 88661 T^{4} + 796645 T^{5} + 6294931 T^{6} + 41663865 T^{7} + 221546909 T^{8} + 813173792 T^{9} + 1804229351 T^{10}$$
$73$ $$1 - 7 T + 276 T^{2} - 984 T^{3} + 30101 T^{4} - 64761 T^{5} + 2197373 T^{6} - 5243736 T^{7} + 107368692 T^{8} - 198787687 T^{9} + 2073071593 T^{10}$$
$79$ $$1 - 18 T + 376 T^{2} - 4711 T^{3} + 57271 T^{4} - 530898 T^{5} + 4524409 T^{6} - 29401351 T^{7} + 185382664 T^{8} - 701101458 T^{9} + 3077056399 T^{10}$$
$83$ $$1 - T + 211 T^{2} - 561 T^{3} + 26112 T^{4} - 69144 T^{5} + 2167296 T^{6} - 3864729 T^{7} + 120647057 T^{8} - 47458321 T^{9} + 3939040643 T^{10}$$
$89$ $$1 - T + 215 T^{2} + 449 T^{3} + 25370 T^{4} + 79560 T^{5} + 2257930 T^{6} + 3556529 T^{7} + 151568335 T^{8} - 62742241 T^{9} + 5584059449 T^{10}$$
$97$ $$1 - 9 T + 214 T^{2} - 2015 T^{3} + 19163 T^{4} - 226496 T^{5} + 1858811 T^{6} - 18959135 T^{7} + 195312022 T^{8} - 796763529 T^{9} + 8587340257 T^{10}$$